PROF. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 2G5 
1 i < - few + _ -VA- _ ■> 
7 — r —— e a [ O'! 2 (1 — l- 2 ) crioRl - »' 2 ) < r /(l - ,,s )i 
(1 — r 3 )* )J 
, f_£/_. - 2j W + & 2 I 
X <3 1 <U 2 (1 - A) 0' 1 cr 2 (l-r 2 ) <r 3 2 (l-J' 2 )i 
_ 1 C ^i 2 __JW_ + ?/3 2 •) 
X e 4 1 O’! 2 (1 - r 2 ) 0-1 o- 2 (1 - r 2 ) tr 2 2 (1 - z' 3 ) I 
X 
or, S denoting summation, since crp = S (x 2 )jn, cr/ = S (y 2 )/?i, the chance varies as 
1 
(1 - r*)- M 
e n 11 
1 — A r-) 
? 
where X is written for S (xy)/(n o^o-g), and S (xy) corresponds to the product-moment 
of dynamics, as S (x 2 ) to the moment of inertia. 
Now, assume r to differ by p from the value previously selected, and expand by 
Taylor’s theorem, after expressing the function, in the following manner : — 
u r — 
e~ n { 
1 —A )•-> 
1 -r*S 
e n [ilog(l -r 2 )- 
1 — \r ■) 
1 
We have 
1 
~ log u rtf = - log u, +' - p + 
(1 + r 2 ) (A - r) 1 X (2r 3 + 6r) - 1 - 6A - 
(1 - ,*? 
, Ji(6 + 36?' 2 + 6 r 4 ) + 4r 5 - 6 U - 28r 3 - 187 ’ a _ 
“r If (1 — r~y d ^ C ' 
Hence log w,. and therefore u r is a maximum when r = X, for the coefficient of p 3 
is then negative. Thus, it appears that the observed result is the most probable, 
when r is given the value S [xyfincr^o-.ffi This value presents no practical difficulty 
in calculation, and therefore we shall adopt it. It is the value given by Bravais, 
but he does not show that it is the best. # 
(c.) Probable Error of the Correlation Coefficients .—Assuming that r has this 
value, we may put X = r in the above result, and we find 
n (1 +r 2 ) p 2 2m-(>- 2 +3) pf 
U r+p = U r e ~ O-rO* 2 ~ (1-rV 3 - ' c - 
Now u r+p is the chance of the observed series on the assumption that the coefficient 
* It seems desirable to draw special attention to this best value of the correlation coefficient, as 
it has hitherto been frequently calculated by methods of somewhat arbitrary character, involving only 
a portion of the observations. 
MDCCCXCVI. —A. 2 M 
